54.3.13 problem 13

Internal problem ID [8569]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 13
Date solved : Wednesday, March 05, 2025 at 06:09:46 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} y^{\prime \prime }+x^{2} y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

Order:=8; 
ode:=diff(diff(y(x),x),x)+x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {x^{4}}{12}\right ) y \left (0\right )+\left (x -\frac {1}{20} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 28

ode=D[y[x],{x,2}]+x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (x-\frac {x^5}{20}\right )+c_1 \left (1-\frac {x^4}{12}\right ) \]

✓ Sympy. Time used: 0.752 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = - \frac {x^{7} r{\left (3 \right )}}{42} + C_{2} \left (\frac {x^{8}}{672} - \frac {x^{4}}{12} + 1\right ) + C_{1} x \left (1 - \frac {x^{4}}{20}\right ) + O\left (x^{8}\right ) \]